A fast regularized least-squares method for retinal

Gunay and Yildirim BioMedical Engineering OnLine 2013, 12:106http://www.biomedical-engineering-online.com/content/12/1/106

RESEARCH Open Access

A fast regularized least-squares method for retinalvascular oxygen tension estimation using aphosphorescence lifetime imaging modelGokhan Gunay1 and Isa Yildirim1,2*

* Correspondence: [email protected] and ElectronicsEngineering Department, IstanbulTechnical University, 34469 Istanbul,Turkey2College of EngineeringDepartment, University of Illinois atChicago, 60607 Chicago, IL, USA

Abstract

Background: Monitoring retinal oxygenation is of primary importance in detectingthe presence of some common eye diseases. To improve the estimation of oxygentension in retinal vessels, regularized least-squares (RLS) method was shown to bevery effective compared with the conventional least-squares (LS) estimation. In thisstudy, we propose an accelerated RLS estimation method for the problem ofassessing the oxygenation of retinal vessels from phosphorescence lifetime images.

Methods: In the previous work, gradient descent algorithms were used to find theminimum of the RLS cost function. This approach is computationally expensive,especially when the oxygen tension map is large. In this study, using a closed-formsolution of the RLS estimation and some inherent properties of the problem at hand,the RLS process is reduced to the weighted averaging of the LS estimates. Thisdecreases the computational complexity of the RLS estimation considerably withoutsacrificing its performance.

Results: Performance analyses are conducted using both real and simulated datasets. In terms of computational complexity, the proposed RLS estimation method issignificantly better than RLS methods that use gradient descent algorithms to findthe minimum of the cost function. Additionally, there is no significant differencebetween the estimates acquired by the proposed and conventional RLS estimationmethods.

Conclusion: The proposed RLS estimation method for computing the retinal oxygentension is computationally efficient, and produces estimates with negligibledifference from those obtained by iterative RLS methods. Further, the results of thisstudy can be applied to other lifetime imaging problems that have similar properties.

Keywords: Accelerated estimation, Closed-form solution, Regularized estimation,Retinal oxygen tension, Phosphorescence lifetime imaging

BackgroundRetinal tissue requires regular oxygenation to prevent some devastating eye diseases,

such as diabetic retinopathy, glaucoma, and age-related macular degeneration [1,2].

Oxygen tension in retinal vessels should therefore be monitored for use in the early

diagnosis of these devastating eye diseases.

Oxygen sensitive microelectrodes can be used to obtain retinal oxygenation [3].

However, this is an invasive method, as the electrodes damage the microenvironment

© 2013 Gunay and Yildirim; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

mailto:[email protected]

http://creativecommons.org/licenses/by/2.0

Gunay and Yildirim BioMedical Engineering OnLine 2013, 12:106 Page 2 of 13http://www.biomedical-engineering-online.com/content/12/1/106

of the retina. Magnetic resonance imaging, a noninvasive imaging technique, has been

used to measure retinal oxygen tension [4]. This approach is limited by its low reso-

lution compared with optical imaging methods.

The oxygen tension of retinal vessels can also be measured using a phosphorescence

lifetime imaging model (PLIM) [5-7]. PLIM is a noninvasive method, though it requires

intravenous injection, and does not damage the microenvironment of the retina. The

least-squares (LS) estimation method has been used to estimate the oxygenation of

retinal vessels based on PLIM. However, the LS method is not robust against noise

contamination, producing high variance and artificial peaks in the estimates, as well as

values outside of the physiological range, which can be attributed to the no use of a

suitable prior model for the data.

To overcome these shortcomings of the LS method, a regularized LS (RLS) estima-

tion method was proposed [8]. The RLS method was developed by utilizing knowledge

of the prior distribution of the model parameters. By applying the RLS method to simu-

lated data as well as real image data acquired from rat retinas, the method was shown

to be robust to the presence of noise, and generated estimates that were more in the

physiologically expected range and whose variance was lower than that obtained with

the LS method which does not use a suitable prior model for the data.

Successful applications of regularization in image processing [9], biomedical engineer-

ing and imaging [10-13], and astronomical imaging [14] motivated the development of

an RLS estimation method for oxygen tension in retinal blood vessels using PLIM [8].

In [8], the regularization window was chosen as a 3×3 window, equivalent to 15x15

μm2. The choice of this size of the window was inspired by the fact that variation

among oxygen tension values in a neighborhood of this size of window should physio-

logically be negligible [15]. The regularization term in the RLS cost function was devel-

oped using this physiological information and assuming that the mean of a pixel in an

oxygen tension map of retinal blood vessels is equal to the sample mean of oxygen

tension values in neighboring pixels. The RLS estimation method was shown to be

superior to the conventional LS estimation method as mentioned before. However, the

gradient-based iterative methods used to find the minimum of the RLS cost function

are computationally costly, especially when considering large oxygen tension maps.

Existing applications have a typical oxygen tension map frame size of around 512 ×

160 pixels. In the early diagnosis of some eye diseases, analyses of small capillaries are

of great importance and require imaging systems with higher resolutions and larger

frame sizes. However, as the map frame size becomes larger, the computational cost of

implementing the RLS estimation method increases drastically. Therefore, to realize

RLS estimation for large oxygen tension maps, faster estimation methods are needed.

In the literature, fast regularized estimation methods have been proposed. For in-

stance, Toh and Yun proposed an accelerated proximal gradient method to minimize a

non-smooth convex regularized cost function [16], and Lampe and Voss proposed a

fast algorithm for Tikhonov-based regularized total LS estimation problems [17].

Mastronardi, Lemmerling and Van Huffel provided a fast regularized total LS algorithm

for solving the basic deconvolution problem [18].

In this study, we propose a computationally efficient RLS method for estimating retinal

oxygen tension using a closed-form solution. The proposed method is shown to be much

faster than the iterative solution, and generates almost identical results. Additionally, the


approach derived in this study can be applied to other imaging problems using PLIM or

FLIM with some minor changes.

Past RLS estimation methodLS estimation of retinal oxygen tension

The Stern–Volmer equation, giving the relationship between the oxygen tension pO2

and the phosphorescence lifetime τ, is as follows:

τ0τ¼ 1þ Kφτ0 pO2ð Þ; ð1Þ

where pO2 (in millimeters of mercury) denotes oxygen tension, Kϕ is the quenching

constant, and τ0 denotes the lifetime in a zero oxygen environment. Based on Eq. (1),

we need to estimate τ to find the value of pO2. The phosphorescence lifetime τ can be

calculated from the distribution of phosphorescence intensity values in different modu-

lation phase images, because the intensity depends on the phase angle, denoted by θ,

between the modulated excitation laser light and the emitted phosphorescence [5,6].

The relation between θ and τ is

tan θ ¼ ωτ; ð2Þ

where ω is the modulation frequency. The intensity can be represented as a function

of θn,

I θnð Þ ¼ k Pd½ �þ 12k Pd½ �mmn cos θð Þ cos θnð Þ þ sin θð Þ sin θnð Þð Þ n¼1; 2; …; np ð3Þ

where mn, θn, and np denote the modulation profile of the image intensifier, the

phase of the gain modulation, and the number of phosphorescence intensity observa-

tions at each pixel location for different phase values of θn, respectively. The concen-

tration of the probe [Pd], k, and the modulation m are unavailable.

If we define a0 = k[Pd], a1 = (1/2)k[Pd]mmncos(θ), and b1 = (1/2)k[Pd]mmnsin(θ),

we can express Eq. (3) as:

I θnð Þ ¼ a0 þ a1 cos θnð Þ þ b1 sin θnð Þ: ð4Þ

The phase angle θ in Eq. (3) is obtained as:

θ ¼ tan−1 b1=a1ð Þ: ð5Þ

Therefore, using Equations (1), (2), and (5), we are able to acquire values for pO2 at

each observation location. Here it should be stressed that pO2 in the Equation 1 can

also be found using phosphorescence quenching measurement. However, for multiple

and closely spaced lifetime applications, the PLIM method satisfies needs of the

application more [19].

In the presence of additive noise and for multiple observations, Eq. (4) can be written

in matrix–vector form as:


Ii1Ii2⋮⋮Iin

266664

377775¼

1 cos θ1ð Þ sin θ1ð Þ1 cos θ2ð Þ sin θ2ð Þ

⋮⋮

1 cos θnð Þ sin θnð Þ

266664

377775

ai0ai1bi1

264

375 þ

Ni1

Ni2⋮⋮Ni

n

266664

377775

ð6Þ

where i denotes the i-th pixel and n denotes the number of phosphorescence intensity

observations for each pixel. We can rewrite Eq. (6) as follows:

y¼ Axþ n ð7Þ

where y, A, x, and n are the phosphorescence lifetime intensity observations, system

matrix, PLIM parameters, and additive noise, respectively. Clearly, to find the model

parameters a0, a1, and b1, at least three phosphorescence intensity observations must

be obtained at three different gain modulation phases for each pixel, though in practice

we need more than three observations because of noise contamination. The LS

estimate of the model parameters is obtained as follows:

a0 a1 b1h iT

¼ Ql�i; ð8Þ

where Q is the pseudo-inverse of the system matrix in Eq. (7).

RLS estimation of retinal oxygen tension

In [14], the RLS cost function for the parameter a1 was defined as:

f ia1 ¼ ai1−Q 2; :ð Þyi� �2 þ β ai1−�ai1

� �2 ð9Þ

where Q(2,:)yi is the LS estimate of the parameter ai1; Q(2,:) is the second row of the

pseudo-inverse of the system matrix A in Eq. (7), yi is the phosphorescence intensity

observation vector of the i-th pixel, and β is the regularization parameter. �ai1 denotes

the mean value of the parameter to be estimated for the i-th pixel. The global cost

function was defined as:

Fa1 ¼XM1

f ia1 ; ð10Þ

where M denotes the total number of pixels in the image. A gradient-based iterative ap-

proach was then used to find the minimum of the RLS cost function.

Derivation of the proposed solution for the RLS estimation

The RLS cost function for the model parameters a0, a1, and b1 for the i-th pixel is

defined as:

CiRLS ¼ yi þ Axi

�� 22þγkxi−�xik22; ð11Þ

where yi is the noisy observation vector, A is the system matrix, and γ is a

regularization coefficient. In Eq. (11), xi is the parameter vector and �xi is its sample

mean, which are defined as:

xi ¼ ai0 ai1 bi1� �T

and �xi ¼ K i; :ð Þa0 K i; :ð Þa1 K i; :ð Þb1½ �T ; ð12Þ


where a0, a1, and b1 are vectors of the phosphorescence lifetime imaging model param-

eters and K is a weighting matrix defining the relations between these parameters. K is

defined as follows:

K j; kð Þ ¼l if j−kj j ¼ 0p if j−kj j ¼ 1 or Rq if j−kj j ¼ R� 10 otherwise

8>><>>:

ð13Þ

where l, p, and q denote the weights of a pixel with respect to itself, to directly adjacent

pixels, and to cross-adjacent pixels, respectively. These are chosen such that l+4p+4q=1. K

(j,k) denotes the weight coefficient of the k-th pixel on the j-th pixel in the oxygen tension

map, and R denotes the number of rows in the map. Note that K is a sparse positive-

definite symmetric Toeplitz matrix.

From Equations (11) and (12), we see that the cost function for a pixel is dependent

on the values of its neighboring pixels. Therefore, there is no pixel-wise solution, and

the problem must be dealt with globally. The global cost function is written as:

CRLS ¼XMi¼1

CiRLS; ð14Þ

where M is the number of pixels in the oxygen tension map.

For any ℜIxJ, ǁXǁF is called the Frobenius norm in ℜIxJ space, and

∥X∥F ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr XTX� �q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr XXT� �q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXI

i¼1

XJ

j¼1

XijXij;

vuut ð15Þ

where tr denotes the trace of a matrix. The operation tr(XTX) is called the Frobenius

inner product [20]. The global cost function using the Frobenius inner product can

now be written as:

CRLS ¼ ∥Y−AX∥2F þ γ∥X−�X∥2

F : ð16Þ

X and Y in Eq. (16) are defined as:

x ¼a10⋯ai0⋯aM0a11⋯ai1⋯aM1b11⋯bi1⋯bM1

��

��¼ x1… xi… xM

� �and Y ¼

I11⋯Ii1⋯IM1⋮⋯⋮ ⋯⋮

I1S⋯IiS⋯IMS

��

��; ð17Þ

where Iis and S are the s-th phosphorescence intensity observation of the i-th pixel and

the number of observations per pixel, respectively. Given the definition of X, Eqs. (12)

and (16) can be rewritten as follows:

�xi ¼ K i; :ð ÞXT� �T

; ð18Þ

CRLS ¼ ∥Y−AX∥2F þ γ∥X− KXT

� �T∥2F : ð19Þ

For the model parameters, the regularization term in the cost function (19) can be

expanded as:

∥X− KXT� �T

∥2F ¼ ∥a0−Ka0∥2

2 þ ∥a1−Ka1∥22 þ ∥b1−Kb1∥2

2: ð20Þ


The least-squares term in (19) can be rewritten as:

∥Y−AX∥2F ¼ tr YTY

� �−2tr YTAX

� �þ tr XTATAX� �

: ð21Þ

Because ATA is equal to

ATA ¼S 0 00 S=2 00 0 S=2

��

��; ð22Þ

tr(XTATAX) becomes

tr XTATAX� � ¼ SaT0 a0 þ S

2aT1 a1 þ S

2 b1Tb1: ð23Þ

From the definition of the Frobenius inner product, tr(YTAX) can be rewritten as:

tr YTAX� � ¼ tr XYTA

� � ¼ aT0 YTA :; 1ð Þ þ aT1 Y

TA :; 2ð Þ þ bT1 YTA :; 3ð Þ: ð24Þ

Considering the above equations, the global cost function can now be written as:

CRLS ¼ tr YTY� �þ aT0 Sa0−2YTA :; 1ð Þ� �þ

aT1S2 a1−2Y

TA :; 2ð Þ� �þ bT1S2b1−2Y

TA :; 3ð Þ� �þγ ∥a0−Ka0∥2

2 þ ∥a1−Ka1∥22 þ ∥b1−Kb1∥2

2

� �:

ð25Þ

We need the model parameters a1 and b1 to estimate the oxygen tension, because

this is nonlinearly dependent on the ratio of b1 and a1. Taking the gradient of the cost

function with respect to the model parameters and equating to zero, we obtain the RLS

estimates of the model parameters.

∇a1CRLS ¼ Sa1−2YTA :; 2ð Þ� �þ Sβ I−2K þ K 2� �

a1 ¼ 0; ð26Þ

a1−RLS ¼ I þ β I−2K þ K 2� �� −1 2

SYTA :; 2ð Þ

; ð27Þ

∇b1CRLS ¼ Sb1−2YTA :; 3ð Þ� �þ Sβ I−2K þ K2� �

b1 ¼ 0; ð28Þ

b1−RLS ¼ I þ β I−2K þ K2� �� −1 2

SYTA :; 3ð Þ

: ð29Þ

In Eqs. (27) and (29), 2YTA :; 2ð Þ� �and 2YTA :; 3ð Þ� �

are the LS estimates of a1 and b1,
S S
respectively. Therefore, the RLS estimates of a1 and b1 can be written as:

a1−RLS ¼ Iþ β I þ KTK−K−KT� �−1

a1;

ð30Þ

b1−RLS ¼ Iþ β I þ KTK−K−KT� �−1

b1;

ð31Þ

where a1 and b1 are the LS estimates of these parameters. By defining a new matrix L

as:

LMxM ¼ I þ β I þ KTK−K−KT� �

; ð32Þ

where I is the identity matrix, the RLS estimates of a1 and b1 can be rewritten in a sim-

pler form as:

a1−RLS ¼ L−1a1; ð33Þ


b1−RLS ¼ L−1b1: ð34Þ

The matrix K in Eq. (32) is a symmetric Toeplitz and positive-definite sparse matrix.

Therefore, L is a symmetric positive-definite matrix. However, it is not in Toeplitz form be-

cause the matrix KTK is not in Toeplitz form. The elements of KTK can be written as:

KTK� �

i j ¼XMk¼1

Ki kKk j ¼XMk¼1

Ki kK j k : ð35Þ

Because K(j,k) = 0 if |j − k| > R + 1, all non-zero elements are within the range 2R+2,

where R is the number of rows in the oxygen tension map. Thus, (KTK)i j = 0 if |i −j| >2R + 2.

If we ignore the first and last 2R+2 rows of the matrix KTK, it is a positive-definite

symmetric Toeplitz matrix. Additionally, as all matrices in L are in Toeplitz form ex-

cept for KTK, the matrix L has the same properties as KTK. That is, except for the first

and last 2R+2 rows, it is a purely positive-definite symmetric Toeplitz matrix. The in-

verse of L shares the same properties as L. Therefore, if we choose a row between the

first and last 2R+2 rows of the inverse of L, we can reconstruct it with negligible errors

in the first and last 2R+2 rows.

Further, even if we change the size of L, there occurs only a negligible difference in

the most significant elements of its inverse. This makes the matrix L and its inverse

very useful when large pO2 images are being used. For a 640 × 480 pO2 map, L is a

307200 × 307200 matrix whose inverse cannot be calculated using a regular computer.

In such cases, iterative approaches must be employed to find the optimum points of

the RLS cost function, and these are also computationally expensive. Because the most

significant elements in L−1 do not vary considerably when its dimension is reduced, we

can calculate a small L−1s and acquire its most significant units, then extend the solution

to larger pO2 maps with negligible errors.

The proposed method is visualized in Figure 1 as a flow chart: (A) Using the phos-

phorescence lifetime images, (B-C) the LS estimates of the model parameters a1 and b1

Figure 1 Flowchart of the proposed method.


of the original pO2 map are found in matrix form. (D) L−1s , whose dimension is Rs2 × Rs

2,

is found for a small Rs × Rs pO2 map, and (E) its middle row is chosen. Rs is chosen as

an odd number to ensure symmetry with respect to the middle unit in the chosen row.

Because L−1s is in Toeplitz form except for the first and last two rows, all other rows

can be recovered exactly using this row with appropriate shifts. Additionally, elements

in the middle row that determine the weights to obtain approximate RLS estimates of

the model parameters (F) can be reshaped to give an Rs × Rs weighted averaging win-

dow. (G) Finally, this weighted averaging window is applied to the LS estimates of the

a1 and b1 parameter maps to find approximate RLS estimates of the model parameters.

Simulated and experimental data results

In the RLS method, the standard Newton–Raphson method is employed to find the

minimum of the RLS cost function. The LS estimates are used as the initial values of

the model parameters to be estimated. In the proposed RLS method, we first determine

L−1s as a 169 × 169 matrix for a 13 × 13 pO2 map. We then follow the procedure given

in the previous section. The computer system used for these simulations is a standard

notebook with an Intel Core Duo 2.27 GHz processor and 4 GB memory. Unless other-

wise stated, the pre-set values of β, window size, and signal-to-noise ratio (SNR) are 5,

13 × 13, and 20 dB, respectively.

We use both simulated and real oxygen tension maps (Figures 2 and 3, respectively)

in our experiments. The real retinal oxygen tension map used in this study was ac-

quired from a Long Evans pigmented rat (~500 g) using a novel system described in

[6]. The animal was treated according to the ARVO Statement for the Use of Animals

in Ophthalmic and Vision Research. In order to anesthetize the rat, an intra-peritoneal

infusion of Ketamine (85 mg/kg IP) and Xylazine (3.5 mg/kg IP) at the respective rates

of 0.5 mg/kg/min and 0.02 mg/kg/min was used. Before and during the imaging, gas

mixture containing 21% oxygen (room air, normoxia) was administered to rats for 5 mi-

nutes. Pd-porphine (Frontier Scientific, Logan, Utah), an oxygen-sensitive molecular

Figure 2 Simulated pO2 map (485 × 600 pixels). Color bar represents pO2 values in millimetersof mercury.

Figure 3 Oxygen tension map (600 × 425 pixels) of a rat generated using the LS (1), iterative RLS(2), and the proposed RLS (3) methods. The color bar shows oxygen tension in millimeters of mercury.


probe, was dissolved (12 mg/ml) in bovine serum albumin solution (60 mg/ml) and

physiological saline buffered to a pH of 7, and injected intravenously (20 mg/kg). Im-

aging was conducted at areas within two-disk diameters (600 microns) from the edge

of the optic nerve head. For each pixel location, 10 phase-delayed optical section phos-

phorescence lifetime intensity images were acquired with a corresponding phase delay

increments of 74 μs, and then these intensity images were brought together to form

en-face phosphorescence images of the retinal vasculatures in the retinal plane. To gen-

erate the simulated data we used in this work, physiological features and topology of

real retinal vessels were followed based on the previous studies [6-8,15]. Near the optic

disc, which has a diameter nearly 300 microns [21], venous and arterial oxygen tensions

of rat retina vary around 35 mm-Hg and 60 mm-Hg, respectively in the normoxia con-

dition. As getting far from the optic disc, arterial oxygen tension decreases almost

linearly while venous oxygen tension remains almost the same [15]. Since the iterative

method requires a considerable amount of time as size of the pO2 map gets larger, the

simulated pO2 map was generated to be 485x600 pixels. But it should be noted that this

size is not a limitation to the proposed method. In the simulations, we add i.i.d. white

Gaussian noise with 15, 20, and 25 dB SNR to the phosphorescence lifetime images.

We examined the MAE performance of the LS, iterative RLS, and proposed RLS

methods for different regularization coefficient values. As shown in Figure 4, there is a

negligible difference between the MAE of the iterative and proposed RLS estimation

methods. On the other hand, there is a significant difference in computation time for

the iterative and proposed RLS methods (see Figure 5). As mentioned previously, the

Newton–Raphson method is used to minimize the RLS cost functions, and its step size,

for our problem, is as follows:

Figure 4 MAE of the LS (er-ls), iterative RLS (er-rls-it), and proposed RLS estimation(er-rls-pr) methods.


δ ¼ 2 þ 2β lþ4pþ4qð Þð Þ−1 ð36Þ

Thus, the step size decreases for higher values of β, and this results in a proportional

increase in computation time.

The time difference between the two methods would be much more notable if the

pO2 map to be estimated were of higher dimensions. Therefore, it is clear that the pro-

posed method is preferable to the iterative one in the sense of MAE performance ver-

sus computational complexity.

From Figure 6, we can see that there is a small difference between the proposed and

iterative approaches in terms of MAE performance for different sizes of the small pO2

Figure 5 Computation times of the iterative RLS (t-it) and proposed RLS (t-pr) estimation methodsfor different values of β.

Figure 6 MAE of the iterative RLS (er-it), and proposed RLS (er-pr) methods for different sizes ofpO2 maps used in the estimation of L−1s .


maps used in the estimation of L−1s . As their size increases, the MAE of the proposed

method converges to that of the standard iterative RLS method.

In Figure 7, we present the computation times of the iterative and proposed RLS

estimation methods for a real oxygen tension map which was extended from original

form 120x85 pixels to 600x425 pixels. As seen from Figures 4 and 6, the results for

the real and simulated oxygen tension maps are proportional in terms of computa-

tion time.

Figure 8 compares the visual results of the proposed and LS estimation methods

for the frame shown in the rectangle in Figure 1. This illustrates the artifacts of the

LS estimation more clearly. As mentioned in [14], the RLS estimation generates

smoother pO2 maps whose values fall within the physiologically expected range. The

acquisition of the real oxygen tension map of a rat’s retina, shown in Figure 3, is

described in [6]. In Figure 3, we compare the visual results of the proposed RLS,

iterative RLS, and LS estimation methods for the real data. When compared with the

Figure 7 Computation times of iterative RLS and proposed RLS estimation methods for differentvalues of β for a real oxygen tension map.

Figure 8 The rectangular frame in Figure 1(1) and its estimates in the presence of noise with 20 dBSNR using the LS (2) and proposed RLS (3) methods. The color bar shows oxygen tension in millimetersof mercury.


LS result, the iterative and proposed RLS methods generate smoother oxygen tension

maps.

Discussions and conclusionPresently, by use of PLIM method, available oxygen tension maps have frame sizes

around 512x160 pixels. As the imaging instruments become enhanced, this size will

rise considerably. The RLS method implemented in an iterative way can handle

oxygen tension images having this size of frames but as shown in the Simulated and

Experimental Data Results Section, it becomes slower as size of the images gets

larger. The proposed method addresses this problem and can be used even for much

larger oxygen tension maps. On the other hand, the proposed method is limited by

the assumption made when developing the regularization window that variation

among oxygen tension values of retinal vessels within this size of window should be

minimal. For the problems in which this assumption is not valid, the proposed

method can generate unsatisfactory results. In this study, by exploiting intrinsic

properties of the problem, we have developed a fast RLS estimation method that is

applicable to large oxygen tension maps. By comparing the results of the available

and proposed approaches, it was shown that, although there is a minor difference in

estimation performance, the proposed method is significantly faster. Additionally,

the proposed method is not restricted to the problem of oxygen tension estimation,

and is applicable to other problems where a similarly strong relationship exists

between neighboring pixels.

AbbreviationsLS: Least squares; RLS: Regularized least squares; PLIM: Phosphorescence lifetime imaging; FLIM: Fluorescence lifetimeimaging; pO2: Oxygen tension.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsEach Author has contributed substantially to the preparation of the paper and approves of its submission to theJournal. Both authors read and approved the final manuscript.

AcknowledgementWe would like to thank Dr. Mahnaz Shahidi for providing us the experimental data which was obtained from ratretinas in her lab.


Received: 19 June 2013 Accepted: 19 September 2013Published: 16 October 2013

References
1. Alder VA, Su EN, Yu DY, Cringle SJ, Yu PK: Diabetic retinopathy: early functional changes. Clin Exp Pharmacol
Physiol 1997, 24:785–788.2. Berkowitz BA, Kowluru RA, Frank RN, Kern TS, Hohman TC, Prakash M: Subnormal retinal oxygenation response

precedes diabetic-like retinopathy. Invest Ophthalmol Vis Sci 1999, 40:2100–2105.3. Wangsa-Wirawan ND, Linsenmeier RA: Retinal oxygen: fundamental and clinical aspects. Arch Ophthalmology

2003, 121:547–557.4. Ito Y, Berkowitz BA: MR studies of retinal oxygenation. Vision Res 2001, 41:1307–1311.5. Shonat RD, Kight AC: Oxygen tension imaging in the mouse retina. Ann Biomed Eng 2003, 31:1084–1096.6. Shahidi M, Shakoor A, Shonat R, Mori M, Blair NP: A method for measurement of chorio-retinal oxygen tension.

Curr Eye Res 2006, 31:357–366.7. Shahidi M, Wanek J, Blair NP, Mori M: Three-dimensional mapping of chorioretinal vascular oxygen tension in

the rat. Invest Ophthalmol Vis Sci 2009, 50:820–825.8. Yildirim I, Ansari R, Wanek J, Yetik IS, Shahidi M: Regularized estimation of retinal vascular oxygen tension from

phosphorescence images. IEEE Trans Biomed Eng 2009, 56(8):1989–1995.9. Elmoataz A, Lezoray O, Bougleux S: Nonlocal discrete regularization on weighted graphs: a framework for

image and manifold processing. IEEE Trans Image Process 2008, 17:1047–1060.10. Funai AK, Fessler JA, Yeo DTB, Olafsson VT, Noll DC: Regularized field map. Estimation in MRI. IEEE Trans Med

Imaging 2008, 27:1484–1494.11. Zhu W, Wang Y, Deng Y, Yao Y, Barbour RL: A wavelet-based multiresolution regularized least squares

reconstruction approach for optical tomography. IEEE Trans Med Imaging 1997, 16:210–217.12. Tarvainen MP, Ranta-aho PO, Karjalainen PA: An advanced detrending method with application to HRV

analysis. IEEE Trans Biomed Eng 2002, 49:172–175.13. Greensite F, Huiskamp G: An improved method for estimating epicardial potentials from the body surface.

IEEE Trans Biomed Eng 1998, 45:98–104.14. Frazin RA: Tomography of the solar corona, a robust, regularized, positive estimation method. Astrophysical J

2008, 530:1026–1035.15. Shakoor A, Blair NP, Mori M, Shahidi M: Chorioretinal vascular oxygen tension changes in response to light

flicker. Invest Ophthalmol Vis Sci 2006, 47(11):4962–4965.16. Toh KC, Yun S: An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares

problems. Pac J Optimization 2008, 6:615–640.17. Lampe J, Voss H: A fast algorithm for solving regularized total least squares problems. Electron Trans Numerical

Anal 2008, 31:12–24.18. Mastronardi1 N, Lemmerling P, Huffel SV: Fast regularized structured total least squares algorithm for solving

the basic deconvolution problem. Numerical Linear Algebra Appl 2004, 12:201–209.19. Shonat RD, Wachman ES, Niu WH, Koretsky AP, Farkas DL: Near-simultaneous hemoglobin saturation and

oxygen tension maps in mouse brain using an AOTF microscope. Biophys J 1997, 73:1223–1231.20. Horn RA, Johnson CR: Matrix Analysis. Cambridge, UK: Cambridge University Press; 1985. ISBN 978-0-521-38632-6.21. Cohan BE, Pearch AC, Jokelainen PT, Bohr DF: Optic disc imaging in conscious rats and mice. Invest Ophthalmol

Vis Sci 2003, 44:160–163.

doi:10.1186/1475-925X-12-106Cite this article as: Gunay and Yildirim: A fast regularized least-squares method for retinal vascular oxygentension estimation using a phosphorescence lifetime imaging model. BioMedical Engineering OnLine 2013 12:106.

Submit your next manuscript to BioMed Centraland take full advantage of:

• Convenient online submission

• Thorough peer review

• No space constraints or color figure charges

• Immediate publication on acceptance

• Inclusion in PubMed, CAS, Scopus and Google Scholar

• Research which is freely available for redistribution

Submit your manuscript at www.biomedcentral.com/submit

Documents

A fast regularized least-squares method for retinal